Questions tagged [cauchy-schwarz-inequality]

The Cauchy-Schwarz inequality states $|\langle x,y \rangle |\leq ||x||\cdot ||y||.$ Use this tag for questions related to the CS inequality and its applications.

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2
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1answer
118 views

Tight sublinear estimates for a triple partial binomial summation

Is there tight estimates for the following logarithmic summation ($\gamma,\gamma'\in(0,1)$ and $\mu,\mu'>0$) $$\log_2\Bigg(\sum_{t=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{...
1
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1answer
97 views

Tight estimates for binomial summation

Is there tight estimates for the following logarithmic summation ($\gamma\in(0,1)$) $$\ln\Bigg(\sum_{t=\frac{n^{}}2-\gamma n^\gamma}^{\frac{n^{}}2+\gamma n^\gamma}\sum_{\ell=\frac{n^{}}2-\gamma n^\...
3
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2answers
366 views

Good upper bound for a certain sum

Given $\gamma \in [0, 1)$, an integer $N \ge 2$, and a decreasing null sequence of positive numbers $e_1,e_2,\ldots,e_t,\ldots$, I'm interested in estimating the sum $S_N := \sum_{t=1}^N\gamma^t e_{N-...
5
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1answer
150 views

Good upper-bound for $\mathbb E[|X-np|^r]$ where $X \sim \text{Binomial}(n,p)$ and $r \ge 1$

Disclaimer. Question moved from SE. Setup Let $X \sim \text{Binomial}(p, n)$, and $r \ge 1$. Question What is a good upper-bound for $\mathbb E[|X-np|^r]$ ? Solution for small $r$ If $r=2$, then ...
2
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0answers
50 views

Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$

Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try... So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...
1
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1answer
265 views

Is there a tight lower bound for the expectation of the product of two positive valued random variables?

Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$. I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely. ...
0
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1answer
86 views

Inequality involving product-of-minus vs minus-of-product for positive integers

I'm encountering this inequality for dimensionality reduction problem. The simplified form looks as follows: Consider positive integers $a_1$, $a_2$, $b_1$ and $b_2$ where $a_1>b_1$ and $a_2>...
3
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1answer
288 views

A moment inequality

Let $\chi(s)=\int_{0}^{1}x(t)^{s}f(t)dt$, where $x(t)$ and $f(t)$ are real valued continuous functions for $t\in[0,1]$, and $f(t)\geq0$. Is it possible to show that $\left(\chi(0)\chi(2)-\chi(1)^{2}...
2
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1answer
164 views

Is it possible to find $f$ such that : $f$ is absolutely integrable, $f'$ is absolutely integrable and such that $f$ is not $1/2$-Hölder

I am trying to find a function $f: \mathbb{R}^+ \to \mathbb{R}^+$ that fullfils the following conditions $$f \in \mathcal{C}^1(\mathbb{R}^+,\mathbb{R}^+)$$ $$\int_{\mathbb{R}^+} f \in \...
-2
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1answer
223 views

On the Cauchy-Schwarz Inequality for trace function of random matrices

In the deterministic case, for two matrices $A$ and $B$ with appropriate matrices, we know that $$tr((A^{T}B)^{2})\leq tr(A^{T}A)tr(B^{T}B)$$ which is the trace form of Cauchy-Schwarz-Inequality (CSI)....
12
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3answers
473 views

A probabilistic angle inequality

Conjecture: There is a universal constant $c$ such that for any fixed nonzero real vector $q$ of any dimension $n$ and any random vector $p$ of the same dimension $n$ with independent components ...
5
votes
1answer
334 views

An inequality involving a sum of power terms

I am currently working in a problem in Information Theory and I came across a difficult inequality. After many attemps, I simplified the inequality, which now looks at follows. Consider a positive ...
2
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2answers
359 views

An alternative proof of Bayesian Cramer-Rao

My question is: Are there an alternative proof of Cramer-Rao lower bound that does not use Cauchy-Swartz inequality? Let me outline the classical proof and explain why I am interested in this ...
0
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1answer
122 views

Upper bound of $\frac{\sum_i c_ia_ie_i}{\sum_i d_ib_if_i}$?

Let $\sum_i c_i =\sum_i d_i=1$, where $c_i,d_i \ge 0$. Assume that $\frac{\sum_i c_ia_i}{\sum_i d_ib_i} \le \epsilon_1$ and $\frac{\sum_i c_ie_i}{\sum_i d_if_i} \le \epsilon_2$, where $a_i,b_i,e_i,f_i ...
4
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0answers
221 views

Does the Cauchy–Schwarz inequality imply 2-positivity?

Recall the following generalisation of Cauchy–Schwarz. Theorem. Let $f\colon \mathscr{A} \to \mathscr{B}$ be a linear 2-positive map between C$^*$-algebras. Then for all $a,b \in \mathscr{A}$ we ...
2
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1answer
127 views

Majorization of cyclic products

Let $k,m,n\in\mathbb N$ such that $n>k$. For a partition $\alpha=(\alpha_1,\dots,\alpha_k)\vdash m$ with $\alpha_1\ge\dots\ge \alpha_k>0$ and nonnegative $ x_1,\dots,x_n$ define $x^\alpha :=\...
2
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2answers
381 views

Does this simple inequality have a name?

Let $x_{1},\ldots,x_{n}$ be nonnegative numbers such that $m \leq x_{i} \leq M$. Let $$ S=\sum_{i=1}^{n}{x_{i}} $$ and $$ Q=\sum_{i=1}^{n}{x_{i}^{2}}. $$ Then $$ Q \leq S(M+m)-nMm. $$ This has ...
5
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0answers
1k views

A stronger Cauchy-Schwarz inequality for traces of compression matrices

Assume that $A$ and $B$ are contractions, so $I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let $C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that: $$Tr\left(\frac{1}{1-AA^T}\right)...
4
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1answer
211 views

A homogeneous but slightly asymmetric inequality

I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$ \begin{equation} \left|\left|\sum_{j=1}^l z_j\right|^p-\sum_{j=1}^l\left|z_j\right|^...
6
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3answers
2k views

Cauchy-Schwarz inequality for bilinear forms valued in an abstract vector space

I previously posted this question on Math.SE but didn't receive an answer. It is perhaps a little vague; part of what I want to know is what question I should ask. First, consider the following form ...
29
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1answer
2k views

Can the fact that the square of an integer is a natural number be categorified?

If $a$ and $b$ are natural numbers, then $a-b$ is an integer and so the square $(a-b)^2$ is a natural number. In particular $$ (a-b)^2 \geq 0. \qquad (1)$$ Combining this fact with the identity $...
2
votes
3answers
573 views

An Linear Algebra Inequality

How to prove the following inequality: Let $X$ and $Y$ be $n\times m$ matrices with real entries. Prove that \begin{equation} \det\left(XY^T\right)^2 \leq \det\left(XX^T\right)\det\left(YY^T\right) . \...
8
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5answers
2k views

A plausible inequality

I come across the following problem in my study. Let $x_i, y_i\in \mathbb{R}, i=1,2,\cdots,n$ with $\sum\limits_{i=1}^nx_i^2=\sum\limits_{i=1}^ny_i^2=1$, and $a_1\ge a_2\ge \cdots \ge a_n>0 $. Is ...